Z Score Calculator
What is Z Score Calculator?
A Z-Score Calculator is a statistical tool that standardizes raw scores by converting them into z-scores using the formula z = (x - μ) / σ. The z-score represents how many standard deviations a data point is from the population mean. This standardization allows for comparison between different datasets and helps determine the relative position of a value within a normal distribution.
The calculator also computes percentiles and probabilities associated with z-scores, making it invaluable for statistical analysis, hypothesis testing, quality control, and educational assessment. It's widely used in psychology, education, finance, and research to identify outliers, compare performance across different scales, and make data-driven decisions based on standardized measurements.
How It Works
Select Type
Choose z-score or raw score calculation
Input Data
Enter score, mean, and standard deviation
Get Analysis
View z-score, percentile, and interpretation
Common Examples
| Scenario | Raw Score | Mean | Std Dev | Z-Score |
|---|---|---|---|---|
| SAT Score | 1200 | 1050 | 200 | 0.75 |
| IQ Test | 115 | 100 | 15 | 1.00 |
| Height (cm) | 180 | 170 | 8 | 1.25 |
| Test Grade | 85 | 78 | 12 | 0.58 |
| Blood Pressure | 140 | 120 | 15 | 1.33 |
Calculation Table
| Z-Score Range | Percentile | Interpretation | Frequency |
|---|---|---|---|
| z = 0 | 50th | Exactly average | Most common |
| z = ±1 | 16th/84th | 1 std dev from mean | 68% within range |
| z = ±2 | 2nd/98th | 2 std dev from mean | 95% within range |
| z = ±3 | 0.1st/99.9th | 3 std dev from mean | 99.7% within range |
Frequently Asked Questions
What does a z-score tell me?
A z-score indicates how many standard deviations a value is from the mean. Positive z-scores are above average, negative are below average, and z=0 is exactly average.
What is considered a high or low z-score?
Z-scores beyond ±2 are considered unusual (less than 5% of data), and beyond ±3 are rare (less than 0.3% of data). Most data falls within ±2 standard deviations.
How do I interpret percentiles?
A percentile shows what percentage of the population scores below that value. For example, the 84th percentile means 84% of people scored lower than that value.
When should I use z-scores?
Use z-scores to compare values from different datasets, identify outliers, standardize test scores, or determine probabilities in normal distributions.
What assumptions does z-score calculation make?
Z-scores assume the data follows a normal distribution. For non-normal data, z-scores may not accurately represent probabilities and percentiles.